It's an interesting question as to how to the bijection between the measurement eigenvalues $\{+1,-1\}$ and the bits $\{0,1\}$ or the qubits $\{|0\rangle,|1\rangle\}$ are reflected or intuited in the mind's eye. Briefly perhaps theoretical computer scientists most often think in terms of $\{0,1\}$ while physicists intuit the measurement as $\{\pm 1\}$.
Similarly consider a computer scientist and her friend the computer engineer. The computer scientist probably intuits bits 0 and 1, while the computer engineer maybe worries about the voltages on various nodes of a transistor. Clearly they can nonetheless communicate with each other, but one just has to be careful in the transliteration.
Indeed imagine understanding Shor's algorithm by considering $Z$-basis measurements of the period as being the eigenvalues of the $Z$-operator. I for one would find that explanation awkward.
That's not to say that algorithms that map 0 to +1 and 1 to -1 can't be useful - indeed, Ryan O'Donnell - a theoretical computer scientist, not a physicist - wrote a whole book on the analysis of Boolean functions, wherein thinking of bitsthe outputs of such functions as +1 and -1 (as opposed to 0 or 1) often provides the most natural and intuitive perspective. For example, this enables perhaps "the right" way to take Fourier transforms of Boolean functions, even classically in the context of various algorithms in learning theory.